Algebraic Geometry and Commutative Algebra (eBook)

In Honor of Masayoshi Nagata

Hiroaki Hijikata, Heisuke Hironaka, Masaki Maruyama (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
406 Seiten
Elsevier Science (Verlag)
978-1-4832-6505-6 (ISBN)

Infos

Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata presents a collection of papers on algebraic geometry and commutative algebra in honor of Masayoshi Nagata for his significant contributions to commutative algebra. Topics covered range from Weierstrass models and endomorphism algebras of abelian varieties to the generic Torelli theorem for hypersurfaces in compact irreducible hermitian symmetric spaces. Coarse moduli spaces for curves are also discussed, along with discriminants of curves of genus 2 and arithmetic surfaces. Comprised of 14 chapters, this volume begins by describing a basic fibration as a Weierstrass model, with emphasis on elliptic threefolds with a section. The reader is then introduced to canonical bundles of analytic surfaces of class VII0 with curves; Lifting Problem on ideal-adically complete noetherian rings; and the canonical ring of a curve. Subsequent chapters deal with algebraic surfaces for regular systems of weights; elementary transformations of algebraic vector bundles; the irreducibility of the first differential equation of Painlevé; and F-pure normal rings of dimension two. The book concludes with an assessment of the existence of some curves. This monograph will be a useful resource for practitioners and researchers in algebra and geometry.

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Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata presents a collection of papers on algebraic geometry and commutative algebra in honor of Masayoshi Nagata for his significant contributions to commutative algebra. Topics covered range from Weierstrass models and endomorphism algebras of abelian varieties to the generic Torelli theorem for hypersurfaces in compact irreducible hermitian symmetric spaces. Coarse moduli spaces for curves are also discussed, along with discriminants of curves of genus 2 and arithmetic surfaces. Comprised of 14 chapters, this volume begins by describing a basic fibration as a Weierstrass model, with emphasis on elliptic threefolds with a section. The reader is then introduced to canonical bundles of analytic surfaces of class VII0 with curves; Lifting Problem on ideal-adically complete noetherian rings; and the canonical ring of a curve. Subsequent chapters deal with algebraic surfaces for regular systems of weights; elementary transformations of algebraic vector bundles; the irreducibility of the first differential equation of Painleve; and F-pure normal rings of dimension two. The book concludes with an assessment of the existence of some curves. This monograph will be a useful resource for practitioners and researchers in algebra and geometry.

Inhalt

Front Cover 1
Algebraic Geometry and Commutative Algebra: In Honor of Masayoshi NAGATA 4
Copyright Page 5
Table of Contents 6
Table of Contents of Volume I 7
Chapter 1. On Weierstrass Models 8
Introduction 8
Notation 8
0. Variation of Hodge structures 9
1. Elementary properties 12
2. Elliptic fibrations 13
3. Elliptic threefolds 22
References 29
Appendix 30
1. Lemmas on ramifications 31
2. Main Theorem 32
References 34
Chapter 2. Canonical Bundles of Analytic Surfaces of Class VII 36
Introduction 36
Notation and Convention 37
1. 37
2. 39
3. 43
4. 44
5. 46
6. 47
References 54
Chapter 3. Ideal-adic Completion of Noetherian Rings II 56
Introduction 56
Notation 58
1. Rotthaus' Hilfssatz. (cf. [19], [11]) 58
2. Proof of Theorem A 64
3. Proof of Theorem B 66
References 69
Chapter 4. Endomorphism Algebras of Abelian Varieties 72
Introduction 72
1. Notations 74
2. Elliptic curves 79
3. Some general facts. Types I and II 82
4. Type III 91
5. Type IV 94
6. Abelian surfaces 97
7. The case dim X = g, a prime number g > 2
8. Survey of the results. Some questions 100
References 102
Chapter 5. On the Canonical Ring of a Curve 106
1. Introduction 106
2. Clifford Index and Green's Conjecture 107
3. Stability Properties of E 110
4. Locally Decomposable Sections of ^dE 116
References 119
Chapter 6. Algebraic Surfaces for Regular Systems of Weights 120
Abstract 120
Contents 120
1. Introduction 120
2. System of weights, having one negative exponent 124
3. System of weights, having one negative exponent and some 0 exponents 168
4. System of weights, whose smallest exponent e is equal to —2 173
5. Weighted homogeneous singularity of dimension two 188
Appendix 198
Appendix A: Rational elliptic surfaces for D4, E6, E7 and E8 199
References 215
Chapter 7. Generic Torelli Theorem for Hypersurfaces in Compact Irreducible Hermitian Symmetric Spaces 218
0. Introduction 218
1. Preliminaries 220
2. Vanishing theorems for the cohomology groups Hq(Y,Opy(k)) 227
3. Jacobian rings and the duality theorem 234
4. Infinitesimal variation of Hodge structure of hypersurfaces 244
5. Polynomial structure 251
6. Symmetrizer lemma 252
7. Moduli space and the period map 255
8. Generic Torelli theorem 257
9. Appendix. (Proof of theorem (2.3.1)) 260
References 266
Chapter 8. A Variety Which Contains a P1-fiber Space as an Ample Divisor 268
Introduction 268
Conventions and notations 269
1. Separably uniruled varieties 270
2. Uniruled varieties with UR(X) = n and non-zero sections in H0(X,Sm(^nOX) ) 276
3. Fiber structure of uniruled varieties 279
4. Variety whose hyperplane section is a P1-bundle 283
References 293
Chapter 9. How Coarse the Coarse Moduli Spaces for Curves Are! 296
0. Introduction 296
1. Hyperelliptic curves with an automorphism of order p 297
2. Kodaira-Spencer map for a hyperellliptic curve 303
3. Main results 310
References 314
Chapter 10. Elementary Transformations of Algebraic Vector Bundles II 316
0. Introduction 316
1. A construction method of algebraic vector bundles 317
2. Some properties of E(Z,W1,...,Wr) 328
3. An example (Horrocks-Mumford bundle) 341
References 350
Chapter 11. Discriminants of Curves of Genus 2 and Arithmetic Surfaces 352
0. Introduction 352
1. A pencil of curves of genus 1 354
2. Pencils of curves of genus 2 356
3. Anaytic theory of pencils of curves of genus 2 361
4. Arithmetic surfaces 365
References 372
Chapter 12. On the Irreducibility of the First Differential Equation of Painlevé 374
1. Review of the preceding paper 375
2. Proof of the Theorem 377
References 391
Chapter 13. Study of F-purity in Demension Two 394
Introduction 394
0. Preliminaries and notations 395
1. Gorenstein case 396
2. Examples of rational singularities which are not of F-pure type 399
References 402
Chapter 14. A Note on the Existence of Some Curves 404
1. Introduction 404
2. An inequality 404
3. Proofs 406
References 407

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-6505-6 / 1483265056
• ISBN-13: 978-1-4832-6505-6 / 9781483265056